What exactly do you mean by "axiom"? A first order sentence? A universally quantified equation?
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GoldsternDec 10 '11 at 22:35

@CSstudent: The equations about complement(a) are automatically true in any Boolean lattice. Did you perhaps mean that some of the ingredients in these equations should be interpreted in the given lattice rather than in the desired Boolean lattice? If so, which ingredients? If not, how should the Boolean lattice be related to the given one? Depending on what the question is supposed to mean, it might be useful to point out that the sublattices of Boolean lattices are exactly the distributive lattices.
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Andreas BlassDec 11 '11 at 1:19

@Andreas: Thanks, all the ingredients of the Boolean lattice should be interpreted in the given lattice. Alternatively, I can formulate the question as: Given a complete lattice, under which condition is lub{b | b /\ a = bottom} = glb{b | b \/ a = top}? With an added distributivity axiom, i.e. a /\ (b \/ c) = (a /\ b) \/ (a /\ c) the lattice becomes Boolean. Is this enough to make the desired equation hold?
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CSstudentDec 11 '11 at 9:16

1 Answer
1

Since you are asking about universally quantified equations, there is only the trivial answer: $x=y$.

Theorem: Assume that $\varphi$ is a (finite or infinite) list of universally quantified equations (sometimes called "laws" or "identities") such that
every complete lattice satisfying $\varphi$ will have complements.
Then there is no nontrivial lattice satisfying $\varphi$, or in other words, the equations $\varphi$ imply $(\forall x,y): x=y$.

Proof: From a nontrivial lattice satisfying $\varphi$ I will construct a complete lattice satisfying $\varphi$ which is not Boolean.